Fractions¶
Recall (doc/postdoc). Not a first course.
Job: field of fractions \(\mathbb{Q}=\operatorname{Frac}(\mathbb{Z})\); template for localization and rational functions.
Core¶
Equivalence: \(\frac{a}{b}\sim\frac{c}{d}\) iff \(ad=bc\) (\(b,d\neq0\)). Reduced form: \(\gcd(|a|,|b|)=1\), \(b>0\).
| Op | Formula |
|---|---|
| \(+/-\) | \(\dfrac{a}{b}\pm\dfrac{c}{d}=\dfrac{ad\pm bc}{bd}\) |
| \(\times\) | \(\dfrac{a}{b}\cdot\dfrac{c}{d}=\dfrac{ac}{bd}\) |
| \(\div\) | \(\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}\) (\(c\neq0\)) |
Field axioms: \((\mathbb{Q},+,\cdot)\) is the prime field of characteristic \(0\).
Egyptian / partial fractions later: decompose in \(\mathbb{Q}(x)\) or over number fields — same algebra, polynomial denominators.
Used later for¶
- Rational expressions: \(K(x)=\operatorname{Frac}(K[x])\)
- Rings and fields: localization \(S^{-1}R\)
- Number theory: Farey, Diophantine approximation
- Measure / probability: ratios of measures as densities
Check¶
Why is \(\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}\) not the field sum?
That is mediant (Farey), not \(+\) in \(\operatorname{Frac}(\mathbb{Z})\); denominators must clear via \(bd\).